Week 1 Task 5: Citizen science project - signs of dangers

The Crevasse, by Edgar Mueller

Today your mission is…

To start doing citizen science with us!

Ready, Set, Go

Review the list of signs or precursors that point to developing dislike and avoidance of mathematics. If these signs describe your child, don’t panic, but take prophylactic measures. The rest of this course is full of activities you can do to help!

Send us stories, examples, and anecdotes that illustrate these dangers. You can talk about your own experiences, or describe your child, your student, a friend, or anyone else you know.

Why do it?

Most grown-ups have “grief stories” when it comes to mathematics. If this happened to you, what if grief comes to the next generation, too? Can you shield your children from harm?

What you see below is our first attempt at putting together a systematic, research-based, people-approved set of answers to that question. We invite you to contribute.

Ask a question if an item below seems relevant, but does not quite make sense. This is work in progress, and your questions will help.

Dangers specific to multiplication

Additive misconceptions

Frequent mistakes by using addition instead of multiplication

Mental blocks about “very multiplicative” ideas, such as proportions, ratios, division of fractions, or combinatorics

36 Replies

1. Ability attributes - for high school I attended a grammar school in the UK, which you needed to pass an examination to go to. I was the only one from my primary school to go there, everyone else went to one of the two local comprehensive mixed ability secondary (high) schools. At primary school I excelled at Maths. I raced my friends through the series of Maths cards, which we used rather than books, and was one of the few who completed the whole series before leaving the school. I asked for homework in Maths regularly and never struggled much with it from what I remember. I found it fun and exciting.

At secondary school we were put into ability sets after the first year, and I was put in one of the two equal status "bottom" sets for this subject. It was a small school, with only 4 ability sets in our year group, and in reality if I had gone to a different school I would have probably been placed in one of the top ability classes, however, as an adolescent this isn't something that occurs to you. I went through the whole of high school believing I was bad at Maths, got very low marks, and didn't enjoy it, which looking back seems really odd considering how much I loved it in my earlier childhood. At the beginning of my last year at the school, the year we took our GCSEs (leaving certificates) I got over 90% in a test, rediscovered my love of mathematics, and ended up getting an A grade for the GCSEs. I guess this is a long winded way of saying that being placed in group perceived as a "low ability" class, my confidence was destroyed and I believed my ability was gone and I assume that I must have stopped putting in the effort to learn that I had done previously until that confidence returned.

2. Can't see connections between math and other endeavours - after regaining my confidence in Maths and rediscovering my love for it, I studied it at a higher level for a while at college and then as a subsidiary at university, but was never shown any practical connection to a future career (other than Maths teacher or accountant!) or application of the subject, so I saw little point in continuing to study it. Later in life I worked in a job where I organized events for high school students to show them future university options, and what they could do with the different subjects that they were studying. I observed the speakers at the Maths day with fascination, suddenly seeing all the opportunities that studying Maths could open up, and how many applications it has, and wished so much that I had had such career / informative advice when I was at that age.

In my opinion schools spend far too much time getting people to learn the abstract concepts (times tables for example - an example of your "mnemonic complexes set in stone" point) and they neglect to inject any life or reality into the subject.

An issue I need to address with my child right now is an inability to understand 0 properly. I teach my children to count objects, not to count by rote. My child now believes that 20 is "10" that follows up the twenty-something's, 30 follows the thirty-something's etc. I would be interested in any other thoughts you have about exactly what kind of misunderstanding this is and how you would work against it.

@ChristyM, your kid seems correct in the essence of the system, but uses an original way to symbolize. Many kids make up pet names for ideas that are new to them. So do all mathematicians, scientists, engineers, artists...

You COULD call the ten that follows twenty-somethings "twenty". You just need to make the rest of your naming system consistent. I am very curious about your kids' system - he's probably onto something! Let's make a little research.

Print out some visual hundred charts that do not have labels. Here is the elegant version our @yelenam made, color-coded for primes:

You can find many more 100-charts in the Week 3 activity called Factorization Diagrams. Say that you and Maria D. are very interested in your child's naming system, and ask to name numbers on the chart. Maybe not all, but enough to see how the system works. Focus on tens. Ask what comes after the last number in this chart (in our terms, ninety-nine). In other words, investigate the system your kid made up!

I took my time to read through all the activities and this one really struck a chord.

Personally, inspite of scoring well in all my classes, I still feel mathematically challenged as I think it was my memory that helped me through. Now, well into professional life, I dont "remember" any of the theorems or proofs. Basically, I dont think I ever appreciated or assimilated the concepts that I learnt. So again, during personal finance planning, some concepts, however simple seem complex :(

For my children, I exactly want the opposite. I want them to be able to appreciate the concept and internalize the beauty of playing with numbers.

Thanks for this list - separating it out makes it easier to think through! My 7-y-o daughter said last week she was tired of getting 100% on her math tests because the other kids made (unkind) comments about it. I wanted to shout "NO! Shine on!" or something because this was such a familiar story to my middle school and early high school years (it's not cool to be good at math). amightygirl.com has some great resources on this as well.

I have say homeschooling has opened my eyes to education.(even as a preschool teacher I was blind to things) I could pass a math class and I had math through 12th grade pre or calculus. I wanted my kids to really understand math. I still struggle with the multiplication tables.

I've taken a couple of days to mull this over. My daughter, who is homeschooled by two people who enjoy and are "good at" math, struggles with math anxiety (eyes glaze over, shuts down around math), and I've been thinking all these thoughts through before responding. Part of her struggle would in school be called a learning disability. She's very kinesthetic, and also visual-spacial, so doing calculations on a black and white page get all jumbled up for her, and then this crashes into her perfectionism, and we end up with a dislike of math. We've done mnemonics, visualizing, manipulatives, movies (she used to like BrainPOPJr and we've used Math-U-See in the past, too), games, and plain old worksheets, in color and black and white plus one-on-one guidance. But when I say "Math" she says, "NOOOOOOO!" and gets worried. This shuts down any receptivity to exploration, so we start out on the wrong foot. As I type I realize I should just call it something else, like games or visual puzzles or something. I would love to find more visual (this class has been great) and kinesthetic ways not only to teach math concepts and skills, but also to bridge to paper computations. Celebrating mistakes is also part of what we do, but she just doesn't buy it. Like the other day we were making gluten free donuts in our new donut pan, and both batches came out tasty but not donut shaped. She cried! I was like - no, this is fine, it's our first attempt, we learned this recipe doesn't work, we learned this other recipe needs tweaking but we're closer, etc - and it just didn't make a difference. She is phobic of making mistakes (despite how she has been taught, I think it's hard wired, and hubby has the same fear, which is both transmitted to daughter - but oddly enough not son as much - and I think is partly just genetic hardwiring) in any feild, and since there is a "right" and a "wrong" in math, she has a lot of fear here.

Thank you for sharing your thoughts, which is sometimes harder if you mull over longer... I think we need to add a caveat about deep perfectionism to the essay about celebrating mistakes. When someone has a serious and systemic issue, as you describe, they need more serious and systemic help than a slogan a-la "Just do it!"

There is an exercise where the purpose is to make as many mistakes as possible in one math problem. With preparation and discussion, maybe you can do something like that? It probably won't be easy, but if you know a task is emotionally charged, you can prepare ahead with extra hugs, kittens in your lap, or whatever else makes you feel better...

The memorization and Mnemonic examples resonate with me. I think that memorization and Mnemonics can really impede number sense, even when they come easily to a child. The memorization and Mnemonics can work so well that you never stop to think about "why" certain things are happening.

I remember going over my multiplication facts and memorizing them (which certainly helped), but I never stepped back to think about the quantities I was dealing with. This never seemed to be a problem in school, but only now do I realize how much I was missing.

"Believes that success is largely due to abilities (rather than effort)" This can be troublesome for both those who believe they have innate ability and those who believe they lack it. I solidly believed this idea for most of my schooling years. I did see some who were capable of doing well because of lots of hard work, but because many things came easily to me I thought that was being successful. It wasn't until I was 16 or 17 and knee deep in AP/honors classes, varsity sports, Jr. engineering team, and all the rest that I realized I had to really apply myself. I was behind my peers in study skills, and it took me through my sophomore year of college to really get caught up in this area. But I failed to take advanced math (beyond calculus or statistics) because I felt overwhelmed, I didn't have the innate ability nor the study skills. My daughter was at the opposite end. Every day in 2nd and 3rd grade she was given timed drills of math facts. Every day she failed to complete any of those sheets. She disliked math, but worse she believed she was not good at it. After a year of homeschooling she had come 180 degrees about math. She loves math! She recognized, in the absence of pressure to remember facts or do dry calculations without context, that she had some ability, but also that she could learn all sorts of math by healthy studying. I am glad that she learned this lesson years before I did.

This all makes a lot of sense to me. I definitely got hooked on mnemonics like the 9 times time table, i would and still subtract one and add the remainder so 9*6 is actually 6-1=5 and 9-5=4 so 54… so bad.

I am confused though, about 'celebrating mistakes' What do you mean by this?

I homeschool my son. He is only 5 right now. But one reason I like it already is that as his mother I know him so well, I know when he is tired, stressed, over worked, bored and silly and so I know when to push a bit further and when to pull back. We have fun and are silly at times, and he likes to make erroneous math sentences and have me correct them, or he likes when I make the mistake and he corrects them. This is fun because I know in a round about way, it confirms for me what he knows and doesn't know, which I am always keeping an unwritten log of. But I also think the goal of mathematics is to measure the world. In my homeschooling we have plenty of room for mistakes and practices, but I do intend for him to be a serious student of the world, and to have reverence and respect for teachers and adults and the thinkers who came before him… I guess I am not sure to what extent or in which context I agree with the idea of celebrating mistakes… Can you, any of you, please tell me more about this?

9*6 is actually 6-1=5 and 9-5=4 so 54 - do you understand WHY this works? If you do, or if you figure it out, then it will turn from a mnemonic into a pattern!

This discussion about mistakes and measuring the world is very deep. Of course you would not celebrate a mistake in building a bridge, or getting the distance to Mars wrong and missing your landing window! There is another, more whimsical and free side to math. Mathematics can be not only about measuring this world where we live, but about making up other, fantastic worlds. In that, mathematics is closer to fairy tales, or science fiction, or fine arts where you can create rather freely. So when I talk about celebrating mistakes, I mean taking them from our world into one of those mathematical Wonderlands.

For example, my math circle kids kept confusing squares and cubes, or circles and spheres. But we played with it, and by now, they talk about three-dimensional squares - or even four-dimensional! From mistakes to abstractions.

"celebrate verb 1) they were celebrating their wedding anniversary: commemorate, observe, mark, keep, honor, remember, memorialize. 2) let's all celebrate! enjoy oneself, have fun, have a good time, have a party, revel, roister, carouse, make merry; informal party, go out on the town, paint the town red, whoop it up, make whoopee, live it up, have a ball. 3) he was celebrated for his achievements: praise, extol, glorify, eulogize, reverence, honor, pay tribute to; formal laud."

I've always been good at maths but one of my biggest problems is related with the way I learnt the multiplication tables (I learnt them by heart). When I try to recall, for example, 8x7 I have to recall the full table to get there and sometimes I have to change the order of numbers, that is I can remember 6x8 but not 8x6 and it makes me feel kind of stupid! I'd love my daughters to learn this in any other way.

I didn't have problems with math at school or at the university, so when I saw the first 'glazed look' on my son's face, I was stunned. As you mentioned in one of your comments here, there is a danger of having it 'too easy', this was my case, so I have to kind of 'repeat my math' now to help my elder son with his math anxiety.

Reading that brought back math anxiety I hadn't thought about in almost two decades. I always felt that I could do better at math if I had enough time to figure out a problem whether it was for a test or for homework. I could never understand why "good at math" was so tied to doing math fast.

I also feel a double stereotype threat about math in that I identify with two demographics that are perceived to be more successful in math--Asian American and scientist. So, not only am I expected to be good at math I am supposed to be incredibly good at math. Which I am not. I was taught math in traditional methods and it never made sense to me and it never came easily to me. When I hear or read, now, about other approaches to learning math I wish it was the way I had first learned it because I think I would have grasped mathematical concepts and ideas a lot better and quicker. I think that is why I am searching for multiple approaches to math to introduce to my child. So, that I can help him find the approach that makes the most sense to him.

I can see several of the psychological maths dangers in my children. My oldest has a bit of a 'I hate maths' complex, stemming I think partly from forced timed tests in school (she is now homeschooled). She does really struggle with her 'math facts', she still needs to count on her fingers too (she is eight). But we are really working on her number sense and 'removing coercion' as suggested in the tips. Yesterday she was figuring out how much money she would need to save for seven of her friends to come to a party at $37 each (for her real need not as an exercise). It did not occur to her to use multiplication, she did repeated adding, but in a (to me) very convoluted way, breaking the numbers apart and adding bits here and bits there. She got to the answer in the end though!

I also give her a lot of input in what she does for maths, as well as providing lots of living maths books which she enjoys. When I see her struggling with something I try and change it up, approach a concept in a different way or take a break from that area.

My younger two (aged 6) are a bit different. One in particular is a real perfectionist and gets very anxious when she is confronted by something she does not recognise. She says she likes to do 'sums', worksheets, but if the problem is written differently she panics. She is also not very fond of open ended questions and explorations, she gets cross and frustrated almost immediately and refuses to participate. On the other hand when she is working on something she has decided she wants to do, she is insistent on getting to the end, even if it involves real struggle on her part. I would really like to involve her in more of the types of maths activities you discuss, but she does not respond well to adult-suggested activities on top of the other issues! So any tips would be most useful...

Her twin at least seems free of maths anxiety for now, and I think will really enjoy the activities.

Uh-oh, I am totally busted. I still, at the age of 38, have to go through large portions of the multiplication tables to answer something like 7x 8. Not sure exactly what is meant by "not using mnemonics" to learn them though, as I always thought most people learned them by essentially "skip counting" their way through. Hope my girls can learn them another way., maybe more securely? I am re learning conceptual math with my girls and find, to my surprise that I do like puzzling over mathematical concepts once I get started, after a lifetime of hating to do equations on paper.

Skip-counting is not mnemonics, it's more a calculation technique. Do you skip-count by 7s or 8s? That's very impressive!

But if you memorize the skip sequence as a song, that's another story. For example, you can sing "four eight twelve sixteen twenty" to the tune of "Old MacDonald had a farm" - and it will help to memorize the skip sequence, but only as a whole. It may interfere with calculations and with algebra in some cases, since you will always need the whole song to get to 20.

Thanks for the clarification, that helps a lot. Yes, I can skip count by sevens and eights but not very far, hence my perennial inability to answer 7x 8 etc. without serious thought. Am planning to finally do this right when I do it with my kids!

If you like to calculate, you can do 8th by doubling three times. It's easier than skip-counting seven times! There is a whole neuroscience-based explanation why 7*8 is the hardest multiplication fact in Dehaene's "The Number Sense." The short version is that both numbers are outside of the subitizing range, outside of the 10- and the 9-patterns, and it's not a full square (which are easier to remember).

7*8 has always been one of the hardest for me to remember. I minored in math and still don't have my multiplication tables completely memorized. What are your thoughts on mnemonic devices like the Times Tales? I've always thought that mnemonic devices like the skip counting songs were good, but have also regarded ones like the 9*9=81 story with suspicion. Would you consider them to be helpful? Dangerous? Harmless?

I consider all non-mathematical mnemonic devices dangerous, when it comes to patters, conjectures, results of computations - anything that has meaning to it. You can use a mnemonic to remember which one is sine and which one cosine, because these are rather random names. But 9*9=81 is not random. It is a part of several neat patterns, and it can be computed, so it's better for our minds to focus on these structures. To memorize times tables, if memory is desired, I would use a spaced repetition tool like Anki. You want your tool to serve you individual facts, straight up (no mnemonics), and only those you actually have trouble remembering. We'll work more on that stuff in Week 4.

Confessing here--I have always had math anxiety–understanding the problem, recognizing problems that I've seen before, and giving up. Had I been in a learning environment that helped me recognize the math in creative things we already where doing (like cutting snowflakes), celebrated mistakes, and allowed me to experience without coercion the math connections in a variety of lessons, e.g. life skills (sewing and cooking), language, science, art and music, and nature my whole-being would have felt more supported. I would have had a more comfortable relationship with math and learning overall!

I have craved more help with "visualizing" scaling, exponential systems. I would love it if a Montessori school with many classrooms (and therefore many sets of materials) would/could combine all their sets of gold bead materials to have a night or weekend when families could come and see firsthand what larger and larger numbers look like. There are logistical problems with this, of course; teachers are rightfully afraid of having their classroom materials get lost.

When I see the word "praise" I cringe because I know what the quick, take-away is for many people. You've explained it well here, however I'd suggest even more emphasis on the fact that the praise needs to be on the effort. I'm posting an article by Alfie Kohn that clarifies for me the differences between "praise" and "encouragement."

Math anxiety: One of the features that had me quickly falling in love with a Montessori environment is that competition is really tamped down. Children may come into the classroom with a built-in competitive edge just by living in our culture, but the teachers or guides help them acclimate to an environment where they aren't to be compared, timed, and tested. Other points I want to share here is the fact that many Montessori "works" are self-correcting allowing the child to correct himself. When a Montessori guide does see a mistake she/he is trained not to swoop in and correct the child, rather she makes note of the error and knows that she needs to present the work again to the child(ren) the following day without talking about the mistakes she saw the day before. (I'm thinking that this is not exactly "celebrating" mistakes, however maybe preferable at the very youngest of ages.)

My own book is intended to help purposefully prepare, at home or classroom, an environment for the very youngest children that scaffolds or builds on what they already know, and generates curiosity and connections (mind hacking if you will), and help to eliminate math anxiety.

"celebrate verb 1) they were celebrating their wedding anniversary: commemorate, observe, mark, keep, honor, remember, memorialize. 2) let's all celebrate! enjoy oneself, have fun, have a good time, have a party, revel, roister, carouse, make merry; informal party, go out on the town, paint the town red, whoop it up, make whoopee, live it up, have a ball. 3) he was celebrated for his achievements: praise, extol, glorify, eulogize, reverence, honor, pay tribute to; formal laud."

I was introduced to modeling or drawing ideas, brainstorming, and lateral thinking as an adult in graphic design classes. Another problem-solving technique suggested was to take a break, sleep on it, let ideas "percolate"; don't be surprised if and when answers come to you when you least expect them (but, only when you've worked on them). Who would have guessed that these same techniques can be applied to mathematics?

My kids are young and don't have any of the math specific misconceptions yet, but I see a few of the math anxiety and stereotype threats. My son, age 7, is a perfectionist and gets very upset when he makes a mistake or thinks something is too hard. I see some of this coming up with my daughter, age 5, as well. We joined a math club but I'm not sure if this has been helpful or hurtful. They seem to like to see the friends but my daughter doesn't want to do the games in there since she sees she is behind the other girls (she is 1-2 yrs younger than most of the class).

My kids are too young to be affected by many of the the higher level thinking problems listed. Instead, I worry that my 3yo daughter may come to feel that math isn't something "she does" because my 5yo son IS so engaged with it. She tries to differentiate herself from her older brother, and since he's very good with building (ie Legos) and math I worry that she will draw away from it. It takes the fun out of it for her when he barges in on an activity and is so much faster and better at it than she is. In fact I have decided to embrace the "girl Legos" because they at least make it quite clear that they are for HER too. She doesn't yet feel that "girls don't do math" -- instead it is more of a sibling rivalry and a difficulty that being the youngest, she is always "behind". I am trying to give her more one-on-one time away from Big Bro, and also to emphasize that these things are What Our Family Does, so they can be part of her identity without competing with her brother. But I would love to hear other advice on this!

Second issue -- my daughter's preschool over-emphasizes writing numerals and doesn't spend enough time with manipulatives. I see her resistance to (and difficulty with) writing numerals getting in the way of developing number sense. The school has many other redeeming qualities, but it is frustrating!

Mmenomic complexes set in stone: yup, this includes myself and my student. I have to count on my fingers sometimes and I run through skip counting to get certain multiplication facts. I've told my daughter specifically the tricks I use to figure out stuff in my head. I realize, they are my tricks for my brain. And, by tricks, I kind of mean it. They take me a while and I feel ridiculous standing there thinking while someone else already got the answer. I work in STEM and I wear a calculator watch. I even feel embarassed writing it. I need to be correct, I'll check it a couple of times and I prefer my staff to double check me. I have said that I don't like to do math in public.

Collapse of scales: I thought this was a universal human trait! We don't see where we fit in time or space very well! :-)

Math anxiety: "I'd rather do anything than math" has been spoken. We definitely can't see the connection between math an other endeavors and can't model or draw concepts. At least, not if you are watching and have mentioned the word math in the past 30 minutes. My daughter will build something or stumble onto a shape playing on those peg boards with rubber bands but as soon as I mention something math-y about it, she doesn't want to hear it.

I guess the way we cope is by having humility. I very clearly remember that feeling of needing more time to get to where everyone else in the class seemed to be. Honestly, at some point my time was better spent doing something else. I struggle with the balance between work on your weaknesses vs. let you strengths shine. I feel like even by writing that I've hinted at that whole "some brains are better at math than others" taboo. If I'm being honest, some brains are better at math than others. We can all do it, we are born puzzle solvers, but why do some of us struggle more?

This is a comprehensive list of problems and I have noticed quite a few of them throughout years of working with students. However, I’d like to focus on psychological and social dangers and among the items in that list, “thinks that some people are born to be good at math and some can’t do it,” is my main focus. I believe this is the most prohibitive factor for my students in learning math concepts. In majority of cases the origin of this “I suck at math” expression comes from home. It kills me when I hear a parent saying, “She/he is like me; math wasn’t my strong subject either.” And I have heard that statement too often!

Another danger, I believe, is the current memorization phobia among both educators and parents. There is no doubt that children need to explore and discover, try and make mistakes, find patterns and connections on their own and in groups. However, I strongly believe that memorization has its own merits and should not be condemned all together. What we should be careful with is what to memorize and how to memorize.

How I’d cope with the problem:

I’d usually focus on confidence building more than on teaching math concepts for a long period of time until I’d feel that the child has a decent perception of his/her own ability. I’d break a concept into very smaller ones and walk my students through small steps with lots of encouragement in each step. We know that negative self-perceptions that come from home with frequent confirmations in school environment grow deep roots in children and it is extremely hard to change them. I just hang on to this hope that by focusing on giving them a more positive perception of their abilities maybe some of them see some light and save themselves.

For memorizing basic facts, I use games and games and games with lots of encouragement.